Abstract

Global rainbow thermometry (GRT) measures the mean size and temperature of an ensemble of spray droplets. The domain of validity of the Airy theory for this technique is established through comparison with Lorenz-Mie theory. The temperature derivation from the inflection points of the Airy rainbow pattern appears to be independent of the type of spray dispersion. Measurements in a water spray are reported. The mean diameter obtained from the rainbow pattern lies between the arithmetic and the Sauter mean diameters measured by phase Doppler anemometry. The temperature measurement by GRT is shown to be accurate within a few degrees Celsius.

© 2003 Optical Society of America

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References

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  1. N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micrometer range,” in 7th International Congress on Optical Methods in Flow and Particle Diagnostics (Sunnyvale, Calif., 1988), pp. 294–304.
  2. S. V. Sankar, K. H. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity and refractive index,” in 3rd International Congress on Optical Particle Sizing (Yokohama, Japan, 1993), pp. 275–284.
  3. J. P. A. J. van Beeck, M. L. Riethmuller, “Simultaneous determination of temperature and size of droplets from rainbow using Airy theory,” in Developments in Laser Techniques and Applications to Fluid Mechanics, R. J. Adrian, D. F. G. Durao, F. Durst, M. V. Heitor, M. Maeda, J. H. Whitelaw, eds. (Springer-Verlag, Berlin, 1994), pp. 330–339.
  4. S. V. Sankar, D. H. Buermann, W. D. Bachalo, “An advanced rainbow signal processor for improved accuracy in droplet temperature measurement,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1996), Vol. 1, pp. 9.3.1–9.3.9.
  5. P. Massoli, “Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets,” Appl. Opt. 37, 3227–3234 (1998).
    [CrossRef]
  6. K. Anders, N. Roth, F. Frohn, “Theoretical and experimental studies of the influence of internal temperature gradients on rainbow refractometry,” in 4th International Congress on Optical Particle Sizing (Nuremberg, Germany, 1995), pp. 419–428.
  7. F. Corbin, A. Garo, G. Gouesbet, G. Grehan, “Réfractomètrie d’arc-en-ciel: application au diagnostic des gouttes avec gradient d’indice,” in Receuil des Actes du 5eme Congrès Francophone de Vélocimétrie Laser (Université et Institut National des Sciences Appliquées of Rouen, Rouen, France, 1996), pp. E1.1–E1.8.
  8. J. P. A. J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of single falling raindrops,” Appl. Opt. 34, 1633–1639 (1995).
    [CrossRef] [PubMed]
  9. J. P. A. J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
    [CrossRef] [PubMed]
  10. J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett. 24, 1696–1698 (1999).
    [CrossRef]
  11. N. Roth, K. Anders, A. Frohn, “Refractive-index measurements for the correction of particle sizing and methods,” Appl. Opt. 30, 4960–4965 (1991).
    [CrossRef] [PubMed]
  12. N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in 6th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1992), pp. 15.5.1–15.5.5.
  13. J. P. A. J. van Beeck, “Rainbow phenomena: development of a laser-based, non-intrusive technique for measuring droplet size, temperature and velocity,” Ph.D. dissertation (Eindhoven University of Technology, Eindhoven, The Netherlands, 1997).
  14. H. C. van de Hulst, Light scattering by small particles (Dover, New York, 1981), pp. 243–246.
  15. G. Gouesbet, G. Gréhan, “Generalized Lorenz-Mie theory for assemblies of spheres and aggregates,” J. Opt. A 1, 706–712 (1999).
    [CrossRef]
  16. X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, G. Gréhan, “Characterization of initial disturbances in liquid jet by rainbow sizing,” Appl. Opt. 37, 8482–8503 (1998).
    [CrossRef]
  17. R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–116 (1991).
    [CrossRef] [PubMed]
  18. R. L. Lee, “Mie theory, Airy theory, and the natural rainbow,” Appl. Opt. 37, 1506–1519 (1998).
    [CrossRef]
  19. J. P. A. J. van Beeck, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
    [CrossRef]
  20. N. Roth, K. Anders, A. Frohn, “Size intensitive rainbow refractometry: theoretical aspects,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1996), Vol. 1, pp. 9.2.1–9.2.6.
  21. C. Tropea, “Developments of specialized phase-Doppler techniques and refractive index measurements,” in Lecture Series on Optical Diagnostics of Particles and Droplets, M. L. Riethmuller, J. P. A. J. van Beeck, eds. (Von Karman Institute, Sint-Genesius-Rhode, Belgium, 1999), pp. 1–29.

2001 (1)

J. P. A. J. van Beeck, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[CrossRef]

1999 (2)

J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett. 24, 1696–1698 (1999).
[CrossRef]

G. Gouesbet, G. Gréhan, “Generalized Lorenz-Mie theory for assemblies of spheres and aggregates,” J. Opt. A 1, 706–712 (1999).
[CrossRef]

1998 (3)

1996 (1)

1995 (1)

1991 (2)

Anders, K.

N. Roth, K. Anders, A. Frohn, “Refractive-index measurements for the correction of particle sizing and methods,” Appl. Opt. 30, 4960–4965 (1991).
[CrossRef] [PubMed]

N. Roth, K. Anders, A. Frohn, “Size intensitive rainbow refractometry: theoretical aspects,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1996), Vol. 1, pp. 9.2.1–9.2.6.

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micrometer range,” in 7th International Congress on Optical Methods in Flow and Particle Diagnostics (Sunnyvale, Calif., 1988), pp. 294–304.

K. Anders, N. Roth, F. Frohn, “Theoretical and experimental studies of the influence of internal temperature gradients on rainbow refractometry,” in 4th International Congress on Optical Particle Sizing (Nuremberg, Germany, 1995), pp. 419–428.

N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in 6th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1992), pp. 15.5.1–15.5.5.

Bachalo, W. D.

S. V. Sankar, D. H. Buermann, W. D. Bachalo, “An advanced rainbow signal processor for improved accuracy in droplet temperature measurement,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1996), Vol. 1, pp. 9.3.1–9.3.9.

S. V. Sankar, K. H. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity and refractive index,” in 3rd International Congress on Optical Particle Sizing (Yokohama, Japan, 1993), pp. 275–284.

Buermann, D. H.

S. V. Sankar, K. H. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity and refractive index,” in 3rd International Congress on Optical Particle Sizing (Yokohama, Japan, 1993), pp. 275–284.

S. V. Sankar, D. H. Buermann, W. D. Bachalo, “An advanced rainbow signal processor for improved accuracy in droplet temperature measurement,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1996), Vol. 1, pp. 9.3.1–9.3.9.

Corbin, F.

X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, G. Gréhan, “Characterization of initial disturbances in liquid jet by rainbow sizing,” Appl. Opt. 37, 8482–8503 (1998).
[CrossRef]

F. Corbin, A. Garo, G. Gouesbet, G. Grehan, “Réfractomètrie d’arc-en-ciel: application au diagnostic des gouttes avec gradient d’indice,” in Receuil des Actes du 5eme Congrès Francophone de Vélocimétrie Laser (Université et Institut National des Sciences Appliquées of Rouen, Rouen, France, 1996), pp. E1.1–E1.8.

Fidrich, M. J.

S. V. Sankar, K. H. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity and refractive index,” in 3rd International Congress on Optical Particle Sizing (Yokohama, Japan, 1993), pp. 275–284.

Frohn, A.

N. Roth, K. Anders, A. Frohn, “Refractive-index measurements for the correction of particle sizing and methods,” Appl. Opt. 30, 4960–4965 (1991).
[CrossRef] [PubMed]

N. Roth, K. Anders, A. Frohn, “Size intensitive rainbow refractometry: theoretical aspects,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1996), Vol. 1, pp. 9.2.1–9.2.6.

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micrometer range,” in 7th International Congress on Optical Methods in Flow and Particle Diagnostics (Sunnyvale, Calif., 1988), pp. 294–304.

N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in 6th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1992), pp. 15.5.1–15.5.5.

Frohn, F.

K. Anders, N. Roth, F. Frohn, “Theoretical and experimental studies of the influence of internal temperature gradients on rainbow refractometry,” in 4th International Congress on Optical Particle Sizing (Nuremberg, Germany, 1995), pp. 419–428.

Garo, A.

F. Corbin, A. Garo, G. Gouesbet, G. Grehan, “Réfractomètrie d’arc-en-ciel: application au diagnostic des gouttes avec gradient d’indice,” in Receuil des Actes du 5eme Congrès Francophone de Vélocimétrie Laser (Université et Institut National des Sciences Appliquées of Rouen, Rouen, France, 1996), pp. E1.1–E1.8.

Giannoulis, D.

Gouesbet, G.

G. Gouesbet, G. Gréhan, “Generalized Lorenz-Mie theory for assemblies of spheres and aggregates,” J. Opt. A 1, 706–712 (1999).
[CrossRef]

X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, G. Gréhan, “Characterization of initial disturbances in liquid jet by rainbow sizing,” Appl. Opt. 37, 8482–8503 (1998).
[CrossRef]

F. Corbin, A. Garo, G. Gouesbet, G. Grehan, “Réfractomètrie d’arc-en-ciel: application au diagnostic des gouttes avec gradient d’indice,” in Receuil des Actes du 5eme Congrès Francophone de Vélocimétrie Laser (Université et Institut National des Sciences Appliquées of Rouen, Rouen, France, 1996), pp. E1.1–E1.8.

Grehan, G.

F. Corbin, A. Garo, G. Gouesbet, G. Grehan, “Réfractomètrie d’arc-en-ciel: application au diagnostic des gouttes avec gradient d’indice,” in Receuil des Actes du 5eme Congrès Francophone de Vélocimétrie Laser (Université et Institut National des Sciences Appliquées of Rouen, Rouen, France, 1996), pp. E1.1–E1.8.

Gréhan, G.

G. Gouesbet, G. Gréhan, “Generalized Lorenz-Mie theory for assemblies of spheres and aggregates,” J. Opt. A 1, 706–712 (1999).
[CrossRef]

X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, G. Gréhan, “Characterization of initial disturbances in liquid jet by rainbow sizing,” Appl. Opt. 37, 8482–8503 (1998).
[CrossRef]

Han, X.

Ibrahim, K. H.

S. V. Sankar, K. H. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity and refractive index,” in 3rd International Congress on Optical Particle Sizing (Yokohama, Japan, 1993), pp. 275–284.

Lee, R. L.

Massoli, P.

Ren, K. F.

Riethmuller, M. L.

J. P. A. J. van Beeck, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[CrossRef]

J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett. 24, 1696–1698 (1999).
[CrossRef]

J. P. A. J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
[CrossRef] [PubMed]

J. P. A. J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of single falling raindrops,” Appl. Opt. 34, 1633–1639 (1995).
[CrossRef] [PubMed]

J. P. A. J. van Beeck, M. L. Riethmuller, “Simultaneous determination of temperature and size of droplets from rainbow using Airy theory,” in Developments in Laser Techniques and Applications to Fluid Mechanics, R. J. Adrian, D. F. G. Durao, F. Durst, M. V. Heitor, M. Maeda, J. H. Whitelaw, eds. (Springer-Verlag, Berlin, 1994), pp. 330–339.

Roth, N.

N. Roth, K. Anders, A. Frohn, “Refractive-index measurements for the correction of particle sizing and methods,” Appl. Opt. 30, 4960–4965 (1991).
[CrossRef] [PubMed]

K. Anders, N. Roth, F. Frohn, “Theoretical and experimental studies of the influence of internal temperature gradients on rainbow refractometry,” in 4th International Congress on Optical Particle Sizing (Nuremberg, Germany, 1995), pp. 419–428.

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micrometer range,” in 7th International Congress on Optical Methods in Flow and Particle Diagnostics (Sunnyvale, Calif., 1988), pp. 294–304.

N. Roth, K. Anders, A. Frohn, “Size intensitive rainbow refractometry: theoretical aspects,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1996), Vol. 1, pp. 9.2.1–9.2.6.

N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in 6th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1992), pp. 15.5.1–15.5.5.

Sankar, S. V.

S. V. Sankar, K. H. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity and refractive index,” in 3rd International Congress on Optical Particle Sizing (Yokohama, Japan, 1993), pp. 275–284.

S. V. Sankar, D. H. Buermann, W. D. Bachalo, “An advanced rainbow signal processor for improved accuracy in droplet temperature measurement,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1996), Vol. 1, pp. 9.3.1–9.3.9.

Tropea, C.

C. Tropea, “Developments of specialized phase-Doppler techniques and refractive index measurements,” in Lecture Series on Optical Diagnostics of Particles and Droplets, M. L. Riethmuller, J. P. A. J. van Beeck, eds. (Von Karman Institute, Sint-Genesius-Rhode, Belgium, 1999), pp. 1–29.

van Beeck, J. P. A. J.

J. P. A. J. van Beeck, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[CrossRef]

J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett. 24, 1696–1698 (1999).
[CrossRef]

J. P. A. J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
[CrossRef] [PubMed]

J. P. A. J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of single falling raindrops,” Appl. Opt. 34, 1633–1639 (1995).
[CrossRef] [PubMed]

J. P. A. J. van Beeck, “Rainbow phenomena: development of a laser-based, non-intrusive technique for measuring droplet size, temperature and velocity,” Ph.D. dissertation (Eindhoven University of Technology, Eindhoven, The Netherlands, 1997).

J. P. A. J. van Beeck, M. L. Riethmuller, “Simultaneous determination of temperature and size of droplets from rainbow using Airy theory,” in Developments in Laser Techniques and Applications to Fluid Mechanics, R. J. Adrian, D. F. G. Durao, F. Durst, M. V. Heitor, M. Maeda, J. H. Whitelaw, eds. (Springer-Verlag, Berlin, 1994), pp. 330–339.

van de Hulst, H. C.

R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–116 (1991).
[CrossRef] [PubMed]

H. C. van de Hulst, Light scattering by small particles (Dover, New York, 1981), pp. 243–246.

Wang, R. T.

Wu, Z.

Zimmer, L.

J. P. A. J. van Beeck, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[CrossRef]

J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett. 24, 1696–1698 (1999).
[CrossRef]

Appl. Opt. (7)

J. Opt. A (1)

G. Gouesbet, G. Gréhan, “Generalized Lorenz-Mie theory for assemblies of spheres and aggregates,” J. Opt. A 1, 706–712 (1999).
[CrossRef]

Opt. Lett. (1)

Part. Part. Syst. Charact. (1)

J. P. A. J. van Beeck, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[CrossRef]

Other (11)

N. Roth, K. Anders, A. Frohn, “Size intensitive rainbow refractometry: theoretical aspects,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1996), Vol. 1, pp. 9.2.1–9.2.6.

C. Tropea, “Developments of specialized phase-Doppler techniques and refractive index measurements,” in Lecture Series on Optical Diagnostics of Particles and Droplets, M. L. Riethmuller, J. P. A. J. van Beeck, eds. (Von Karman Institute, Sint-Genesius-Rhode, Belgium, 1999), pp. 1–29.

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micrometer range,” in 7th International Congress on Optical Methods in Flow and Particle Diagnostics (Sunnyvale, Calif., 1988), pp. 294–304.

S. V. Sankar, K. H. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler/rainbow refractometer system for simultaneous measurement of droplet size, velocity and refractive index,” in 3rd International Congress on Optical Particle Sizing (Yokohama, Japan, 1993), pp. 275–284.

J. P. A. J. van Beeck, M. L. Riethmuller, “Simultaneous determination of temperature and size of droplets from rainbow using Airy theory,” in Developments in Laser Techniques and Applications to Fluid Mechanics, R. J. Adrian, D. F. G. Durao, F. Durst, M. V. Heitor, M. Maeda, J. H. Whitelaw, eds. (Springer-Verlag, Berlin, 1994), pp. 330–339.

S. V. Sankar, D. H. Buermann, W. D. Bachalo, “An advanced rainbow signal processor for improved accuracy in droplet temperature measurement,” in 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1996), Vol. 1, pp. 9.3.1–9.3.9.

K. Anders, N. Roth, F. Frohn, “Theoretical and experimental studies of the influence of internal temperature gradients on rainbow refractometry,” in 4th International Congress on Optical Particle Sizing (Nuremberg, Germany, 1995), pp. 419–428.

F. Corbin, A. Garo, G. Gouesbet, G. Grehan, “Réfractomètrie d’arc-en-ciel: application au diagnostic des gouttes avec gradient d’indice,” in Receuil des Actes du 5eme Congrès Francophone de Vélocimétrie Laser (Université et Institut National des Sciences Appliquées of Rouen, Rouen, France, 1996), pp. E1.1–E1.8.

N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in 6th International Symposium on Applications of Laser Techniques to Fluid Mechanics (Instituto Superior Technico, Lisbon, Portugal, 1992), pp. 15.5.1–15.5.5.

J. P. A. J. van Beeck, “Rainbow phenomena: development of a laser-based, non-intrusive technique for measuring droplet size, temperature and velocity,” Ph.D. dissertation (Eindhoven University of Technology, Eindhoven, The Netherlands, 1997).

H. C. van de Hulst, Light scattering by small particles (Dover, New York, 1981), pp. 243–246.

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Figures (12)

Fig. 1
Fig. 1

Setup for GRT. The water spray, the receiving lens system, a transparent screen, and a video camera can be seen. The laser beam is added during postprocessing of the photograph and does not represent the real laser beam thickness.

Fig. 2
Fig. 2

Typical global rainbow pattern in a water spray recorded by a video camera. Only the so-called Airy fringes are visible. The horizontal axis is proportional to the scattering angle.

Fig. 3
Fig. 3

Typical rainbow pattern coming from a single droplet in a water spray. Note the high-frequency ripple structure superimposed upon the low-frequency Airy fringes.

Fig. 4
Fig. 4

Simulation of global rainbow patterns by the Lorenz-Mie approach for three numbers of droplets N tot at constant mean diameter δ = 50 μm and constant dispersion σ = 0.2.

Fig. 5
Fig. 5

Simulation of global rainbow patterns by Lorenz-Mie (lm) and Airy (ai) approaches for several mean diameters δ at constant dispersion σ = 0.2 and for a total number of droplets N tot fixed at 100.

Fig. 6
Fig. 6

Simulation of global rainbow patterns by Lorenz-Mie (lm) and Airy (ai) approaches for several dispersions σ at constant mean diameter δ = 50 μm and for a total number of droplets N tot fixed at 100.

Fig. 7
Fig. 7

Rainbow diameter, computed from θinf2 - θinf1, as a function of spray dispersion σ for various numbers of droplets N tot [Eq. (7)].

Fig. 8
Fig. 8

Deviation of θrg, based on θinf2 and θinf1, from its value at σ = 0, as a function of σ for three mean diameters δ [Eq. (8)].

Fig. 9
Fig. 9

Experimental versus numerical rainbow patterns in the core of the spray at z = 40 cm from the spray nozzle.

Fig. 10
Fig. 10

Comparison of GRT and PDA. The rainbow, Sauter mean, and arithmetic mean diameters are presented along a radial profile in a flat fan water spray at z = 40 cm from the water spray nozzle; p is nozzle pressure.

Fig. 11
Fig. 11

Comparison of rainbow, Sauter mean, and arithmetic mean diameters for an axial profile; p is the nozzle pressure.

Fig. 12
Fig. 12

Temperature variations measured by GRT in an isothermal spray at two different axial positions; p is the operating pressure.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

fd=1σd2π1/2exp-12lnd/δσ2,
fdiΔdi=1/Ntot.
D10=i=1NtotdifdiΔdi=1Ntoti=1Ntot di,
D32=i=1Ntot di3/i=1Ntot di2.
RnbwAix  i=1NtotAix, di2di7/3,
RnbwLMx  i=1Ntot|Ex, di, zi|2,
Drainbow=531.6λθinf2-θinf1-3/2,
θrg=θinf1-13.91λ/Drainbow2/3,

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